# Properties

 Label 5610i Number of curves $4$ Conductor $5610$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5610.h1")

sage: E.isogeny_class()

## Elliptic curves in class 5610i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5610.h3 5610i1 [1, 1, 0, -52377, -4634859]  18432 $$\Gamma_0(N)$$-optimal
5610.h2 5610i2 [1, 1, 0, -58157, -3556311] [2, 2] 36864
5610.h1 5610i3 [1, 1, 0, -376907, 86267439]  73728
5610.h4 5610i4 [1, 1, 0, 168113, -24237389]  73728

## Rank

sage: E.rank()

The elliptic curves in class 5610i have rank $$1$$.

## Modular form5610.2.a.h

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} + q^{11} - q^{12} + 2q^{13} - q^{15} + q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels. 